2. Show that if seven integers from 1 to 12 are chosen, then two of them...

 

 

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2. Show that if seven integers from 1 to 12 are chosen, then two of them will add up to 13. 3. Let T be an equilateral triangle whose sides are of length 1 unit. Show that if any five points are chosen lying on or inside the triangle, then two of them must be no more than unit apart. 4. Show that if any eight positive integers are chosen, two of them will have the same remainder when divided by 7. 5. Show that if seven colors are used to paint 50 bicycles, at least eight bicycles will be the same color. 6. Ten people volunteer for a three-person committee. Ev- ery possible committee of three that can be formed from these ten names is written on a slip of paper, one slip for each possible committee, and the slips are put in ten hats. Show that at least one hat contains 12 or more slips of paper. 7. Six friends discover that they have a total of $21.61 with them on a trip to the movies. Show that one or more of them must have at least $3.61. 10. Show that there must be at least 90 ways to choose six integers from 1 to 15 so that all the choices have the same sum. 11. How many friends must you have to guarantee at least five of them will have birthdays in the same month? 12. Show that if five points are selected in a square whose sides have length 1 inch, at least two of the points must be no more than 2 inches apart. 13. Let A be an 8x8 Boolean matrix. If the sum of the entries in A is 51, prove that there is a row i and a column j in A such that the entries in row i and in column j add up to more than 13. 14. Write an exercise similar to Exercise 13 for a 12 x 12 Boolean matrix. 15. Prove that if any 14 integers from 1 to 25 are chosen, then one of them is a multiple of another. 16. How large a subset of the integers from 1 to 50 must be chosen to guarantee that one of the numbers in the subset is a multiple of another number in the subset? 17. How large a subset of the integers from 1 to n must be chosen to guarantee that one of the numbers in the subset is a multiple of another number in the subset? 2. Show that if seven integers from 1 to 12 are chosen, then two of them will add up to 13. 3. Let T be an equilateral triangle whose sides are of length 1 unit. Show that if any five points are chosen lying on or inside the triangle, then two of them must be no more than unit apart. 4. Show that if any eight positive integers are chosen, two of them will have the same remainder when divided by 7. 5. Show that if seven colors are used to paint 50 bicycles, at least eight bicycles will be the same color. 6. Ten people volunteer for a three-person committee. Ev- ery possible committee of three that can be formed from these ten names is written on a slip of paper, one slip for each possible committee, and the slips are put in ten hats. Show that at least one hat contains 12 or more slips of paper. 7. Six friends discover that they have a total of $21.61 with them on a trip to the movies. Show that one or more of them must have at least $3.61. 10. Show that there must be at least 90 ways to choose six integers from 1 to 15 so that all the choices have the same sum. 11. How many friends must you have to guarantee at least five of them will have birthdays in the same month? 12. Show that if five points are selected in a square whose sides have length 1 inch, at least two of the points must be no more than 2 inches apart. 13. Let A be an 8x8 Boolean matrix. If the sum of the entries in A is 51, prove that there is a row i and a column j in A such that the entries in row i and in column j add up to more than 13. 14. Write an exercise similar to Exercise 13 for a 12 x 12 Boolean matrix. 15. Prove that if any 14 integers from 1 to 25 are chosen, then one of them is a multiple of another. 16. How large a subset of the integers from 1 to 50 must be chosen to guarantee that one of the numbers in the subset is a multiple of another number in the subset? 17. How large a subset of the integers from 1 to n must be chosen to guarantee that one of the numbers in the subset is a multiple of another number in the subset?